Bilinear Dispersive Estimates Via Space Time Resonances, Dimensions Two and Three

Germain, Pierre

doi:10.1007/s00205-014-0764-7

Cited by 17 publications

(25 citation statements)

References 19 publications

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“…The cubic terms in the right-hand side do not lose derivatives, so such an identity can be used to prove local regularity. However, in our problem, the solution is expected to have strictly less that 1/t pointwise decay (as pointed out in [6]). As a result, an identity like (1.10) cannot be used directly to control the long-term growth of the high order energy.…”

Section: 2mentioning

confidence: 67%

“…However, this optimal pointwise decay cannot be propagated by the nonlinear flow, even in simpler semilinear evolutions, due to the presence of a large set of space-time resonances. This was pointed out by Bernicot-Germain [6] who found a logarithmic loss. In this paper we prove t −1+κ pointwise decay of the nonlinear solution (U e , U b ), for certain κ > 0 small.…”

Section: 2mentioning

confidence: 83%

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The Euler–Maxwell System for Electrons: Global Solutions in 2D

Deng

Ionescu

Pausader

2017

Arch Rational Mech Anal

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A basic model for describing plasma dynamics is given by the Euler-Maxwell system, in which compressible ion and electron fluids interact with their own self-consistent electromagnetic field. In this paper we consider the "one-fluid" Euler-Maxwell model for electrons, in 2 spatial dimensions, and prove global stability of a constant neutral background.In 2 dimensions our global solutions have relatively slow (strictly less than 1/t) pointwise decay and the system has a large (codimension 1) set of quadratic time resonances. The issue in such a situation is to solve the "division problem". To control the solutions we use a combination of improved energy estimates in the Fourier space, an L 2 bound on an oscillatory integral operator, and Fourier analysis of the Duhamel formula. YU DENG, ALEXANDRU D. IONESCU, AND BENOIT PAUSADERGuo-Ionescu-Pausader [23] (small irrotational perturbations of constant solutions), following earlier partial results in simplified models in [22,25,18,30]. See also the introduction of [23] for a longer discussion of the Euler-Maxwell system in 3D, and its connections to many other models in mathematical physics, such as the Euler-Poisson model, the Zakharov system, the KdV, and the NLS.A simplification of the full system is the "one-fluid" model, which accounts for the interaction of electrons and the electromagnetic field, but neglects the dynamics of the ion fluid. Under suitable irrotationality assumptions, this model can be reduced to a coupled system of two Klein-Gordon equations with different speeds and no null structure. While global results are classical in the case of scalar wave and Klein-Gordon equations, see for example [33,34,36,37,38,39,9,10,42,44,14,15,2,3,4,5], it was pointed out by Germain [17] that there are key new difficulties in the case of two Klein-Gordon equations with different speeds. In this case, the classical vector-field method does not seem to work well, and there are large sets of resonances that contribute in the analysis.The one-fluid Euler-Maxwell model in 3D was analyzed in [18], using the "space-time resonance method", and the authors proved global existence and scattering, with weak decay like t −1/2 . A more robust result for this problem, which gives time-integrability of the solution in L ∞ , for all parameters, was obtained by two of the authors in [30].In this paper we consider the one-fluid Euler-Maxwell model 1 in 2D. As in dimension 3, in the irrotational case this can still be reduced to a quasilinear coupled system of two Klein-Gordon equations with different speeds and no null structure. At the analytical level, one has, of course, all the difficulties of the 3D problem, such as large sets of resonances. In addition, there is one critical new difficulty, namely the slow decay of solutions, as it was observed by Bernicot-Germain [6] that the nonlinear solutions cannot have the "almost" integrable 1/t decay, due to strong resonant quadratic interactions. This slow decay and the presence of large sets of resonances require new ideas to contro...

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Section: 2mentioning

confidence: 67%

Section: 2mentioning

confidence: 83%

The Euler–Maxwell System for Electrons: Global Solutions in 2D

Deng

Ionescu

Pausader

2017

Arch Rational Mech Anal

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“…Nonlinear Klein-Gordon equation. By combining (1.12) and (1.13), we see that the field φ satisfies a nonlinear Klein-Gordon equation associated with the unknown curved metric g: 14) which is expected to uniquely determine the evolution of the matter (after prescribing suitable initial data). Throughout, we assume that…”

Section: Dynamics Of Massive Mattermentioning

confidence: 99%

“…Good semi-linear Metric-metric GS hh (p, k) Z I 1 J 1 K 1 ∂ a hZ I 2 J 2 K 2 ∂h interior only: (s 2 /t 2 )∂ I 1 L J 1 ∂ 0 hZ I 2 J 2 K 2 ∂ 0 h Curvature-curvature GS ρ ♯ ρ ♯ (p, k) Z I 1 J 1 K 1 ∂ a ρ ♯ Z I 2 J 2 K 2 ∂ρ ♯ interior only: (s 2 /t 2 )∂ I 1 L J 1 ∂ 0 ρ ♯ ∂ I 2 L J 2 ∂ 0 ρ ♯ Curvature-matter GS ρ ♯ φ (p, k) ∂ I 1 L J 1 ∂ a ρ ♯ Z I 2 J 2 K 2 ∂φ Z I 1 J 1 K 1 ∂ρ ♯ Z I 2 J 2 K 2 ∂ a φ interior only: (s 2 /t 2 )Z I 1 J 1 K 1 ∂ρ ♯ Z I 2 J 2 K 2 ∂ a φ Matter-matter GS φφ (p, k) Z I 1 J 1 K 1 ∂ a φZ I 2 J 2 K 2 ∂φ interior only: (s 2 /t 2 )Z I 1 J 1 K 1 ∂ 0 φZ I 2 J 2 K 2 ∂ 0 φ Table: Quasi-linear and semi-linear terms (with the range of indices omitted) 14 The quasi-null structure for the null-hyperboloidal frame 14.1 The nonlinear wave equations for the metric components…”

Section: Homogeneous Functions and Nonlinearities Of The Field Equationsmentioning

confidence: 99%

The global nonlinear stability of Minkowski space for the Einstein equations in the presence of a massive field

LeFloch

2016

Comptes Rendus. Mathématique

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We study the initial value problem for two fundamental theories of gravity, that is, Einstein's field equations of general relativity and the (fourth-order) field equations of f (R)-modified gravity. For both of these physical theories, we investigate the global dynamics of a self-gravitating massive matter field when an initial data set is prescribed on an asymptotically flat and spacelike hypersurface, provided these data are sufficiently close to data in Minkowski spacetime. Under such conditions, we thus establish the global nonlinear stability of Minkowski spacetime in presence of massive matter. In addition, we provide a rigorous mathematical validation of the f (R)-theory based on analyzing a singular limit problem, when the function f (R) arising in the generalized Hilbert-Einstein functional approaches the scalar curvature function R of the standard Hilbert-Einstein functional. In this limit we prove that f (R)-Cauchy developments converge to Einstein's Cauchy developments in the regime close to Minkowski space.Our proofs rely on a new strategy, introduced here and referred to as the Euclidian-Hyperboloidal Foliation Method (EHFM). This is a major extension of the Hyperboloidal Foliation Method (HFM) which we used earlier for the Einstein-massive field system but for a restricted class of initial data. Here, the data are solely assumed to satisfy an asymptotic flatness condition and be small in a weighted energy norm. These results for matter spacetimes provide a significant extension to the existing stability theory for vacuum spacetimes, developed by Christodoulou and Klainerman and revisited by Lindblad and Rodnianski.The EHFM approach allows us to solve the global existence problem for a broad class of nonlinear systems of coupled wave-Klein-Gordon equations. As far as gravity theories are concerned, we cope with the nonlinear coupling between the matter and the geometry parts of the field equations. Our method does not use Minkowski's scaling field, which is the essential challenge we overcome here in order to encompass wave-Klein-Gordon equations.We introduce a spacetime foliation based on glueing together asymptotically Euclidian hypersurfaces and asymptotically hyperboloidal hypersurfaces. Well-chosen frames of vector fields (null-semi-hyperboloidal frame, Euclidian-hyperboloidal frame) allow us to exhibit the structure of the equations under consideration as well as to analyze the decay of solutions in time and in space. New Sobolev inequalities valid in positive cones and in each domain of our Euclidian-hyperboloidal foliation are established and a bootstrap argument is formulated, which involves a hierarchy of (almost optimal) energy and pointwise bounds and distinguishes between low-and high-order derivatives of the solutions. Dealing with the (fourth-order) f (R)-theory requires a number of additional ideas: a conformal wave gauge, an augmented formulation of the field equations, the quasi-null hyperboloidal structure, and a singular analysis to cope with the limit f (R) → R. 2

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“…Therefore there is an integer l such that |x 3 | 2 ≤ 3|(Ax) l | 2 . With denoting the Japanese bracket, we then have x 3 ≤ 3 (Ax) l and so, x 3 ≤ 3 n k=1,2 (Ax) k ) x k (Ax) 3 , because in all cases t ≥ 1. In terms of the functional E A , this gives,…”

Section: 2mentioning

confidence: 99%

Resonant Hamiltonian systems associated to the one-dimensional nonlinear Schrödinger equation with harmonic trapping

Fennell

2019

Communications in Partial Differential Equations

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We study two resonant Hamiltonian systems on the phase space L 2 (R → C): the quintic one-dimensional continuous resonant equation, and a cubic resonant system that has appeared in the literature as a modified scattering limit for an NLS equation with cigar shaped trap. We prove that these systems approximate the dynamics of the quintic and cubic one-dimensional NLS with harmonic trapping in the small data regime on long times scales. We then pursue a thorough study of the dynamics of the resonant systems themselves. Our central finding is that these resonant equations fit into a larger class of Hamiltonian systems that have many striking dynamical features: non-trivial symmetries such as invariance under the Fourier transform and the flow of the linear Schrödinger equation with harmonic trapping, a robust wellposedness theory, including global wellposedness in L 2 and all higher L 2 Sobolev spaces, and an infinite family of orthogonal, explicit stationary wave solutions in the form of the Hermite functions.

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Bilinear Dispersive Estimates Via Space Time Resonances, Dimensions Two and Three

Cited by 17 publications

References 19 publications

The Euler–Maxwell System for Electrons: Global Solutions in 2D

The Euler–Maxwell System for Electrons: Global Solutions in 2D

The global nonlinear stability of Minkowski space for the Einstein equations in the presence of a massive field

Resonant Hamiltonian systems associated to the one-dimensional nonlinear Schrödinger equation with harmonic trapping

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